Suggestions
Use up and down arrows to review and enter to select.Please wait while we process your payment
If you don't see it, please check your spam folder. Sometimes it can end up there.
If you don't see it, please check your spam folder. Sometimes it can end up there.
Please wait while we process your payment
By signing up you agree to our terms and privacy policy.
Don’t have an account? Subscribe now
Create Your Account
Sign up for your FREE 7-day trial
Already have an account? Log in
Your Email
Choose Your Plan
Individual
Group Discount
Save over 50% with a SparkNotes PLUS Annual Plan!
Purchasing SparkNotes PLUS for a group?
Get Annual Plans at a discount when you buy 2 or more!
Price
$24.99 $18.74 /subscription + tax
Subtotal $37.48 + tax
Save 25% on 2-49 accounts
Save 30% on 50-99 accounts
Want 100 or more? Contact us for a customized plan.
Your Plan
Payment Details
Payment Summary
SparkNotes Plus
You'll be billed after your free trial ends.
7-Day Free Trial
Not Applicable
Renews September 28, 2023 September 21, 2023
Discounts (applied to next billing)
DUE NOW
US $0.00
SNPLUSROCKS20 | 20% Discount
This is not a valid promo code.
Discount Code (one code per order)
SparkNotes PLUS Annual Plan - Group Discount
Qty: 00
SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.
Choose Your Plan
For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!
You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.
Members will be prompted to log in or create an account to redeem their group membership.
Thanks for creating a SparkNotes account! Continue to start your free trial.
Please wait while we process your payment
Your PLUS subscription has expired
Please wait while we process your payment
Please wait while we process your payment
The radius of a sphere is increasing at a rate of 2 meters per second. At what rate is the volume increasing when the radius is equal to 4 meters?
This type of problem is known as a "related rate" problem. In this sort of problem, we know the rate of change of one variable (in this case, the radius) and need to find the rate of change of another variable (in this case, the volume), at a certain point in time (in this case, when r = 4). The reason why such a problem can be solved is that the variables themselves have a certain relation between them that can be used to find the relation between the known rate of change and the unknown rate of change.
Related rate problems can be solved through the following steps:
Step one: Separate "general" and "particular" information.
General information is information contained in the problem that is true at all times.
Particular information is information that is true only at the particular instant that the
problem is asking about.
For example, in this problem, the general information is that the radius is growing at a rate of 2 meters per second. Thus, if we let the radius of the sphere be represented by r, we can say that r'(t)=2 m/s.
The particular information is that at the moment of interest, r(t) = 4.
It is important to separate out these two types of information because information about the particular case must be set aside and must not be used until the very last step of solving the problem. One mistake that students frequently make is to use the particular information too early. This invariably leads to incorrect answers.
Step two: Draw a sketch of the situation using the general information only.
This often helps to illuminate the relationship between the variables.
In this case, the general information leads to the following sketch:
Although we know the rate of change of r, we cannot depict a rate in the sketch.
Note here that it would be a mistake to write "r = 4" in the sketch, since this is only true at one particular instant.
Step three: Identify the known rate and the desired rate
In this case, we know , and we want to find
, the rate of
change of the volume.
Step four: Use geometry or trigonometry to relate the variable with the known
rate of change to the variable with the unknown rate of change.
The appropriate relation will often involve one of the following:
V = ![]() |
V(t) = ![]() ![]() ![]() |
V(t) = ![]() ![]() ![]() | |||
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() | |||
![]() ![]() ![]() | |||
![]() |
Please wait while we process your payment